Algebra & Number Theory

Identifying Frobenius elements in Galois groups

Tim Dokchitser and Vladimir Dokchitser

Full-text: Open access

Abstract

We present a method to determine Frobenius elements in arbitrary Galois extensions of global fields, which may be seen as a generalisation of Euler’s criterion. It is a part of the general question how to compare splitting fields and identify conjugacy classes in Galois groups, which we will discuss as well.

Article information

Source
Algebra Number Theory, Volume 7, Number 6 (2013), 1325-1352.

Dates
Received: 4 August 2011
Revised: 9 May 2012
Accepted: 7 June 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730030

Digital Object Identifier
doi:10.2140/ant.2013.7.1325

Mathematical Reviews number (MathSciNet)
MR3107565

Zentralblatt MATH identifier
1368.11115

Subjects
Primary: 11R32: Galois theory
Secondary: 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 12F10: Separable extensions, Galois theory

Keywords
Frobenius elements Artin representations Galois groups

Citation

Dokchitser, Tim; Dokchitser, Vladimir. Identifying Frobenius elements in Galois groups. Algebra Number Theory 7 (2013), no. 6, 1325--1352. doi:10.2140/ant.2013.7.1325. https://projecteuclid.org/euclid.ant/1513730030


Export citation

References

  • A. R. Booker, “Numerical tests of modularity”, J. Ramanujan Math. Soc. 20:4 (2005), 283–339.
  • W. Bosma, J. Cannon, and C. Playoust, “The Magma algebra system, I: The user language”, J. Symbolic Comput. 24:3-4 (1997), 235–265.
  • J. P. Buhler, Icosahedral Galois representations, Lecture Notes in Mathematics 654, Springer, Berlin, 1978.
  • C. U. Jensen, A. Ledet, and N. Yui, Generic polynomials: Constructive aspects of the inverse Galois problem, Mathematical Sciences Research Institute Publications 45, Cambridge University Press, 2002.
  • D. P. Roberts, “Frobenius classes in alternating groups”, Rocky Mountain J. Math. 34:4 (2004), 1483–1496.
  • D. Zagier, “Elliptic modular forms and their applications”, pp. 1–103 in The 1-2-3 of modular forms, edited by K. Ranestad, Springer, Berlin, 2008.